Breakdown of Ohm's Law by Disorders in Low-Dimensional Transistors

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  The length-dependent transfer curves of ALD In$_2$O$_3$ FETs are useful data, but the paper's central claim, that disorder-induced Anderson localization makes the conductance decay as $\exp(-L/\xi)$ and breaks Ohm's law at room temperature, does not follow from them. What moved the stance is the paper's own millimetre-scale data: digitized, Fig. S3b shows the normalized conductance of 21 devices ($W = 6\,\mu$m, $L = 2$ to $12{,}000\,\mu$m) changing by only 0.12 to 0.56 decades at $V_{GS} = 0.5$ to $2.5$ V, which needs $\xi$ of 11 to 45 mm in the paper's model, while Figs. 2 to 4 report $\xi$ of 0.1 to about 10 $\mu$m; the paper cites this figure as confirming localization. In addition, $\xi$ is defined as a coherence length, and no evidence of micrometre phase coherence at 295 K is given; the length dependence is equally described by a length-dependent threshold voltage, which the model excludes by assumption and no measurement tests; and the scaling-function figure (Fig. 5) restates the fitted exponential form rather than testing the scaling theory. All 31 references exist.

  What is measured and what is claimed

  The paper measures transfer curves of In$_2$O$_3$ FETs (0.8 to 3 nm channels, 5 nm HfO$_2$, Ni contacts) against channel length from 0.04 to 2 $\mu$m (Figs. 2, 3, S4 to S7), against temperature from 10 to 295 K (Fig. 4, S8), and on millimetre devices (Fig. S3). It defines $\sigma_{normal} = I_D L/(V_{DS} W)$, fits $\sigma_{normal} = \sigma_0(V_{GS})\exp(-L/\xi(V_{GS}))$, and states that "the channel-length-dependent behavior arises solely from the localization effect". It interprets $\xi$ as "the characteristic spatial extent over which an electron's wavefunction remains coherent before becoming localized in real space" and calls the result "the compelling evidence for the first experimental observation of electron localization in atomically thin In2O3 oxide semiconductors." The measurements look careful (negligible hysteresis, many lengths); the question is whether they support this interpretation.

  The millimetre devices contradict the exponential length law

  I digitized Fig. S3b from a 300 dpi render of v1, with the axes calibrated from the tick marks (69.4 px per decade, 102 px per volt) and the 21 curves identified by their order, which is monotonic in $L$, and checked against the legend colours. Relative to the 2 $\mu$m device, $\log_{10}\sigma_{normal}$ of the 12,000 $\mu$m device is $-0.56$ at $V_{GS} = 0.5$ V, $-0.39$ at 1.0 V, $-0.19$ at 2.0 V and $-0.12$ at 2.5 V. For $L \le 30\,\mu$m it stays within 0.07 decades of the 2 $\mu$m value at every one of these biases. A least-squares fit of $\ln\sigma_{normal}$ linear in $L$ over all 21 devices gives $\xi = 11$, 16, 33 and 45 mm, and a law linear in $\ln L$ fits 1.8 to 3.4 times better (residual sums 0.54 against 1.84, 0.29 against 0.92, 0.11 against 0.24, 0.08 against 0.14). The largest change is a step between $L = 120$ and 180 $\mu$m (0.14 decades at 1 V), after which $\sigma_{normal}$ stays within 0.08 decades from 180 to 6,000 $\mu$m at 1 V; a step suggests two groups of devices rather than a length law. With the localization lengths the paper reports ($\le$ about 10 $\mu$m), $\exp(-L/\xi)$ would suppress the 12 mm device by $10^{-500}$ or more. The channel thickness and $V_{DS}$ of Fig. S3 are not stated; if $V_{DS}$ is 0.05 or 0.1 V as in Figs. 2 to 4, the 2 $\mu$m device has $\sigma_{normal} \approx 1.3$ to $2.7\times10^{-5}$ S at $V_{GS} \approx 2.9$ V, below $e^2/h$, where the paper's own criterion places strong localization. The supplement states that the similar $V_{th}$ shift "confirms that localization originates from intrinsic channel material properties"; the digitized data show instead that the length effect in these devices is confined to the turn-on region while the on-state scales almost Ohmically.

  Room-temperature localization needs micrometre phase coherence

  In the standard theory the paper builds on (its refs. 5, 6), a conductance that decays as $\exp(-L/\xi)$ describes phase-coherent transport over the length $L$; at finite temperature a sample longer than its phase-coherence or hopping length conducts Ohmically. The paper extracts $\xi$ up to about 10 $\mu$m at 295 K (Fig. 4c) and reads $\xi$ as a coherence length, but gives no measurement of the phase-coherence length (for example weak-localization magnetoresistance) or of hopping, and does not discuss dephasing. The Fig. 4b data at $V_{GS} = 1$ V and 295 K lie at $\ln\sigma_{normal} \approx -8.7$ to $-8.85$ (read from the figure), that is $\sigma \approx 3.7$ to $4.3\,e^2/h$, so by the paper's own relation $k_F l = (h/e^2)\sigma$ (Note S1) $k_F l \approx 4$, on the diffusive side of its Ioffe-Regel criterion, yet a strong-localization exponential is fitted there.

  A length-dependent threshold voltage describes the same data

  The log-scale transfer curves in Figs. 2c, 3a, S4a and S5a look like copies shifted along $V_{GS}$, and the $\xi(V_{GS})$ curves at different temperatures in Fig. 4c look like copies shifted along $V_{GS}$ too. If length only shifts the transfer curve, $\sigma(V_{GS},L) = f(V_{GS} - V_{th}(L))$, then by the chain rule $1/\xi = -\partial\ln\sigma/\partial L = (\partial\ln\sigma/\partial V_{GS})\,dV_{th}/dL$ at every $L$. Above threshold, where $\sigma \propto V_{GS} - V_{th}$, this gives $\xi \propto V_{GS} - V_{th}$, a straight line in $V_{GS}$, which is the shape of Fig. 2f; and the length dependence fades as $\sigma$ rises well above threshold, which is what the paper attributes to the Ioffe-Regel crossover at $e^2/h$. So the extracted $\xi$ may carry no information beyond one number $dV_{th}/dL$ and the transfer curve. The paper's model assumes $V_{th}$ independent of $L$ and does not test this alternative, nor classical mechanisms that shift $V_{th}$ with length: finite-size percolation in the random potential of the film (the paper cites percolation work, refs. 7, 26, 30, but does not discuss it as an alternative; Tseng et al., ACS Nano 20, 11756 (2026), published after this v1, attribute length- and width-dependent $V_T$ in amorphous In$_2$O$_3$ to it), or doping near the contacts. A shift of $V_{th}$ with $L$ is a real and important observation; attributing it to Anderson localization is the step the evidence does not carry.

  The scaling-function figure restates the fit

  Fig. 5 is presented as "the first direct experimental realization of the full scaling function $\beta$". For $\sigma = \sigma_0\exp(-L/\xi)$, $\beta = d\ln\sigma/d\ln L = -L/\xi = \ln\sigma - \ln\sigma_0$ identically (checked with exactory-derive: consistent), so any data that follow the fitted exponential with a weakly varying $\sigma_0$ fall on a line of slope 1 in $\beta$ against $\ln\sigma$, whatever the mechanism. The figure also covers only the $\beta < 0$ branch, so it is not the full function, and it is at 10 K, not at the room temperature where the main claims are made.

  Internal consistency and reporting

  Fig. 4b and Fig. 4c disagree on the order of the 60 K and 250 K data at $V_{GS} = 1$ V: in Fig. 4b the 60 K line has the second smallest slope ($\xi \approx 6\,\mu$m from its end points), in Fig. 4c the 60 K curve has the second shortest $\xi$ (about 2.5 $\mu$m) and 250 K a longer one (about 5 $\mu$m); a legend or colour swap is likely. The channel width of the Fig. 2 and 3 devices, the thickness, anneal and $V_{DS}$ of the Fig. S3 devices, the $V_{th}$ extraction method, and the device layout (whether different lengths come from one die) are not given, which limits the length analysis. The printed DOI of ref. 14 lacks a hyphen (10.1038/s41563019-0455-8; the record is 10.1038/s41563-019-0455-8).

  References

  I checked all 31 references with exactory-check lookup: the 30 with DOIs resolve to registry records (author-count warnings came from my abbreviated author lists), and ref. 21 (Hu et al., IEDM 2022) resolves on Crossref from its printed DOI. The references are real; this does not bear on the stance.

  What decided the stance

  The claims-follow-from-evidence criterion. The data establish a channel-length-dependent threshold voltage in ultrathin In$_2$O$_3$ FETs. They do not establish Anderson localization at room temperature: the millimetre data in the supplement contradict the exponential law with the reported $\xi$, no coherence evidence is given, an alternative that fits the same curves is excluded by assumption, and the scaling-function figure is not an independent test.

  • consistencyminor

    Details needed to analyse the length dependence are missing: the channel width of the Fig. 2 and 3 devices, the thickness, anneal and $V_{DS}$ of the Fig. S3 devices, the $V_{th}$ extraction method, and whether devices of different length share a die. Fig. S3b shows a step between L = 120 and 180 $\mu$m that such information would explain or exclude.

    • https://arxiv.org/abs/2601.01283v1· Methods; Figs. 2, 3, S3
  • claimsminor

    Fig. 5's slope of 1 in $\beta = d\ln\sigma/d\ln L$ against $\ln\sigma$ follows from the fitted form: for $\sigma = \sigma_0 e^{-L/\xi}$, $\beta = -L/\xi = \ln\sigma - \ln\sigma_0$. It is not an independent test of the scaling theory, and it covers only the $\beta < 0$ branch at 10 K, not 'the full scaling function'.

    • https://arxiv.org/abs/2601.01283v1· Fig. 5 and section 'Scaling Theory of Electron Localization'— Identity checked with exactory-derive (status consistent).
  • methodsubstantive

    The model fixes $V_{th}$ independent of L ('arises solely from the localization effect'), yet a rigid length-dependent threshold shift gives $1/\xi = (\partial\ln\sigma/\partial V_{GS})\,dV_{th}/dL$ exactly and $\xi \propto V_{GS} - V_{th}$ above threshold, the linear shape of Fig. 2f; no measurement separates the two, and classical causes of a length-dependent $V_{th}$ (finite-size percolation, near-contact doping) are not discussed.

    • https://arxiv.org/abs/2601.01283v1· Eq. in Fig. 1c; Figs. 2c, 2f, 4c
    • https://doi.org/10.1021/acsnano.5c21838· Section 2.4— Later work (April 2026) attributing length- and width-dependent V_T in amorphous In2O3 to finite-size percolation.
  • referencescitation check: upheld

    Abrahams et al. (1979), the scaling theory the paper tests, exists as cited.

    Evidence · citation_lookup/2.0.0

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  • consistencyminor

    Fig. 4b and Fig. 4c disagree on the ordering of the 60 K and 250 K data at $V_{GS}$ = 1 V (Fig. 4b: 60 K slope second smallest, $\xi \approx 6\,\mu$m; Fig. 4c: 60 K $\xi \approx 2.5\,\mu$m, second shortest).

    • https://arxiv.org/abs/2601.01283v1· Fig. 4b, 4c
  • referencescitation check: upheld

    Nenashev et al. (2019), the percolation description of AOS transport cited as ref. 30, exists as cited.

    Evidence · citation_lookup/2.0.0

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  • evidencesubstantive

    Digitized Fig. S3b (W = 6 $\mu$m, 21 devices, L = 2 to 12,000 $\mu$m): relative to L = 2 $\mu$m, $\log_{10}\sigma_{normal}$ of the 12,000 $\mu$m device is $-0.56$, $-0.39$, $-0.19$, $-0.12$ at $V_{GS}$ = 0.5, 1.0, 2.0, 2.5 V, and within 0.07 decades for all L up to 30 $\mu$m. The exponential law then needs $\xi$ = 11 to 45 mm, against the 0.1 to about 10 $\mu$m reported in Figs. 2 to 4, although the supplement cites Fig. S3 as confirming localization.

    • https://arxiv.org/abs/2601.01283v1· Fig. S3b and its caption; Figs. 2f, 3e, 4c— Digitized from a 300 dpi render; axis calibration from tick marks; curves identified by order and legend colour.
  • claimssubstantive

    $\xi$ is read as a coherence length ('remains coherent before becoming localized'), and values up to about 10 $\mu$m are extracted at 295 K, but the paper gives no measurement or estimate of the phase-coherence length or hopping length at room temperature, so the Anderson-localization interpretation of the room-temperature length dependence is not supported.

    • https://arxiv.org/abs/2601.01283v1· Section 'Electron Localization in In2O3 FETs'; Fig. 4c
  • referencesminor

    The printed DOI of ref. 14 (10.1038/s41563019-0455-8) does not resolve; the record is 10.1038/s41563-019-0455-8.

    • https://doi.org/10.1038/s41563-019-0455-8· ref. 14
  • evidenceminor

    At $V_{GS}$ = 1 V and 295 K (Fig. 4b), $\sigma_{normal} \approx$ 1.4 to 1.7 $\times 10^{-4}$ S (read from the figure), about 4 $e^2/h$, so $k_F l \approx 4$ by the paper's Note S1, on the diffusive side of its own criterion, while a strong-localization exponential is fitted.

    • https://arxiv.org/abs/2601.01283v1· Fig. 4b; Note S1
  • referencescitation check: upheld

    Das Sarma and Hwang (2014), cited for the Ioffe-Regel crossover, exists as cited.

    Evidence · citation_lookup/2.0.0

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  Impact prediction: top 25% of 1,791 Physics, Mesoscale and Nanoscale Physics papers, 2025-07-01 to 2025-12-31

  top 1%

  What to do next

  Next step on this line

  Separate a length-dependent threshold voltage from localization

  Ground

  The extracted $\xi$ equals $(\partial\ln\sigma/\partial V_{GS})^{-1}(dV_{th}/dL)^{-1}$ if length only shifts the transfer curve, and the millimetre devices scale almost Ohmically in the on-state.

  Action

  Extract $V_{th}(L)$ and check whether one $dV_{th}/dL$ per film reproduces $\xi(V_{GS}, T)$ in Figs. 2 to 4; add 10 to 100 $\mu$m devices on the same die and a magnetoresistance measurement of the phase-coherence length.

  Expected outcome

  Either the transfer curves change shape with L beyond a shift and $\sigma$ decays exponentially over lengths well beyond $\xi$ on the same film, which would support localization, or the length effect reduces to $V_{th}(L)$, whose cause can then be studied directly.

  A different direction

  Model the length and width dependence as finite-size percolation with measured scales

  Ground

  Amorphous oxide channels have a random potential landscape with measured amplitude (STS) and correlation length (microwave impedance microscopy), and classical percolation needs no phase coherence at room temperature.

  Action

  Build a random-resistor model with those independently measured scales and predict $V_{th}(L, W)$ and $\sigma_{normal}(L, V_{GS}, T)$, including the millimetre devices and devices below 100 nm.

  Expected outcome

  A quantitative fit with no free length scale would identify the mechanism and give design rules for scaled BEOL oxide transistors; a failure would narrow the room left for localization.

  Would change this verdict: I would move to sound with (a) devices of 10 to 100 $\mu$m on the same films as Figs. 2 to 4 showing $\sigma_{normal}$ suppressed as $\exp(-L/\xi)$ with the reported $\xi$, together with an account of Fig. S3b; (b) an independent measurement of phase coherence over micrometres at room temperature, such as weak-localization magnetoresistance; and (c) a test that separates localization from a rigid shift of $V_{th}$ with $L$, for example a change of subthreshold slope or transfer-curve shape with $L$ that no shift can reproduce, or the temperature dependence of $dV_{th}/dL$.